An important theme in modern number theory is that of \(p\) -variation—the idea
of interpolating a family of mathematical objects or invariants by a \(p\) -adic parameter.
For instance, Iwasawa constructed an “arithmetic” \(p\) -adic \(L\) -function, which interpolates
the \(p\) -part of class groups of a \(\mathbb {Z}_p\) -tower over a number field.
The existence (and finiteness) of this object allows one to deduce the remarkable
Iwasawa class number formula (see e.g. [1]}).
A general principle is as follows: if \(M\)  is a motive over a number field \(K\) 
and \(\rho :G_K \rightarrow \mathbb {Z}_p\)  is a additive character, we may consider a family
of motives \(M \otimes \rho _\chi \) . Here \(\rho _\chi \)  denotes the composition \(\chi \circ \rho \)  and \(\chi \)  varies over the space \(\mathrm {Hom}(\mathbb {Z}_p,p^\times )\)  of continuous characters.
One expects that \(p\) -adic families of invariants can be constructed that interpolate
classical invariants.
For example, we may consider a motive \(M_f\)  corresponding to an eigenform \(f\) .
The algebraic special values of \(L(M_f \otimes \chi ,s)^\mathrm {alg}\) , where \(\chi \)  is a Dirichlet character of \(p\) -power degree,
can be parameterized by a \(p\) -adic \(L\) -function (see e.g. [2]}). This is somewhat miraculous, as it tells us that \(L(M_f\otimes \chi ,1)^\mathrm {alg}\) 
is close to \(L(M_f\otimes \chi ^{\prime }, 1)^\mathrm {alg}\)  \(p\) -adically if \(\chi \)  and \(\chi ^{\prime }\)  are sufficiently close.
